Nano-architected metamaterials: Carbon nanotube-based nanotrusses

Carbon 131 (2018) 38e46

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Nano-architected metamaterials: Carbon nanotube-based nanotrusses Chunyi Zhang a, b, Abdolhamid Akbarzadeh b, c, *, Wei Kang a, d, Jianxiang Wang a, e, Armin Mirabolghasemi b a

HEDPS, Center for Applied Physics and Technology, College of Engineering, Peking University, Beijing 100871, China Department of Bioresource Engineering, McGill University, Island of Montreal, QC H9X 3V9, Canada c Department of Mechanical Engineering, McGill University, Montreal, QC H3A 0C3, Canada d Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China e State Key Laboratory for Turbulence and Complex System, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 December 2017 Received in revised form 19 January 2018 Accepted 20 January 2018

In this paper, we propose a novel face-centered cubic (fcc) lattice-like nanotruss with carbon nanotubes and newly designed 12-terminal junctions as building blocks. Using molecular dynamics simulation, both thermal and mechanical properties of this novel nano-architected metamaterials are systematically predicted and a comparison study is conducted between fcc and simple cubic lattice-like nanotrusses. Our findings demonstrate that the fcc lattice-like nanotrusses have relatively low thermal conductivity k ¼ 1:24  3:31W=ðm,KÞ and reasonably high specific modulus M ¼ 49:79  114:15MN,m=kg in all three directions ([100], [110], [111]), for a material density in the range of r ¼ 0:23  0:82g=cm3 . Depending on the specific requirements in material design, the widely ranged Young's modulus E and thermal conductivity k of nano-architected metamaterials can be tuned by changing the tube length. All the aforementioned advantages can be maintained at a high temperature up to 3000K, which indicates fcc lattice-like nanotrusses are superior to most existing mechanically robust thermal insulators. To showcase the scale independency of the unprecedented properties of the proposed nano-architected metamaterials, thermal and mechanical properties of their macroscopic counterparts are studied using standard mechanics multiscale homogenization. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Low-dimensional nanomaterials, like one-dimensional (1D) carbon nanotubes (CNTs) [1] and two-dimensional (2D) graphenes [2], have attracted considerable attention from material scientists and engineers, owning to their outstanding features, such as extremely high thermal conductivity [3e6], excellent mechanical strength [7e9], chirality-dependent electrical conductivity [10e12], and large specific surface area [13]. These unprecedented properties, however, are only observed along specific directions, e.g. tube axial direction for CNTs and plane direction for graphenes. 3D carbon nanotube networks [14], which employ various

* Corresponding author. Department of Bioresource Engineering, McGill University, Island of Montreal, QC H9X 3V9, Canada. E-mail address: [email protected] (A. Akbarzadeh). URL: http://www.mcgill.ca/bioeng/faculty-and-staff/abdolhamid-akbarzadehshafaroudi https://doi.org/10.1016/j.carbon.2018.01.082 0008-6223/© 2018 Elsevier Ltd. All rights reserved.

junctions [15] to link CNTs covalently, are emerging as one of the most promising lightweight multifunctional materials and have recently attracted considerable research interest, since they are aligned with the growing research efforts on architected cellular solids [16] and cellular-based metamaterials [17]. Earlier studies have reported various types of junctions, including the 3-terminal Y- and T-junctions [18e20], 4-terminal X-junction [21,22], and a few 3D multi-terminal junctions [23,24]. Some studies have also implemented fullerene-based junctions to assemble carbon nanotubes [25]. These junctions have been successfully synthesized in experiments utilizing various approaches: electron or ion beam welding [20,26,27], chemical vapor deposition [28], arc discharge [29] and nanochannel alumina [30]. Assembling single-walled carbon nanotubes (SWCNTs) by the junctions, different types of carbon nanotube networks can be developed, such as super carbon nanotubes [31], stacked hexagonal superarchitectures [32], supercubic nanotrusses, and superdiamond nanotrusses [32e34]. These CNT networks not only extend some excellent properties of low-

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dimensional nanomaterials to three dimensions but also possess many extraordinary properties which their constitutive building blocks do not possess. For instance, 3D CNT networks are one of the best candidates for hydrogen storage due to their low density, large surface area, and continuous porosity [35e37]. The unique thermal properties and electrical rectifying effect make these nanotube networks a promising material for next-generation thermal management [38,39] and electronic [40e42] nanodevices, respectively. In addition, 3D CNT networks can be applied as strain sensors due to their durability and high sensitivity to strain [43]. Inspired by the ceramic nanolattice structures introduced by Meza et al. [44], we propose a novel face-centered cubic (fcc) lattice-like hollow nanotruss which consists of 12-terminal junctions and (8,8) armchair SWCNTs. In order to synthesize a Yjunction, Terrones et al. [45] removed one of the “arms” of an Xjunction by using careful conditions of irradiation, which indicates that three, four, or even more terminals can be generated with controlled electron irradiation. Despite the fact that several studies have investigated both thermal and mechanical properties of simple cubic (sc) and body-centered cubic (bcc) lattice-like nanotrusses [23,25,33,34], there is a lack of knowledge about thermal and mechanical properties of the CNT-based fcc latticelike nanotruss. To the best of our knowledge, the only recent paper studied the fcc lattice-like structure [46] is based on graphenes rather than nanotubes, which built the nanotrusses in a different way from our proposed model; the paper merely discussed mechanical properties and focused on compression tests. The unitcell length in the aforementioned study was fixed at 5.5 nm, in which the properties of the nanotruss are still dominated by the junctions rather than the architecture. In this work, we construct fcc lattice-like nanotruss based on (8,8) single-walled carbon nanotubes. Using molecular dynamics (MD) simulation, both thermal and mechanical properties (including thermal conductivity, Young's modulus, specific modulus, initial fracture strain, initial fracture strength, and ultimate strength) of fcc and sc lattice-like nanotrusses are obtained. For convenience, these two nanotrusses are termed as SC and FCC hereinafter, owing to their similarity to the simple cubic (sc) and face-centered cubic (fcc) crystal structures, respectively. We focus on the architecture rather than the junction configuration, and the length of SWCNTs is changed over a wide range. Tensile tests are conducted in [100], [110] and [111] directions. Thermal and mechanical properties of the FCC nanotruss are compared with the SC nanotruss. The observation demonstrates that FCC nanotrusses can be employed as lightweight and mechanically robust thermal insulators with tunable Young's modulus, thermal conductivity and high thermal stability. 2. Model and computational method 2.1. Model Unit cells of the SC and FCC nanotrusses are illustrated in Fig. 1a and f, respectively. We covalently join the (8,8) SWCNTs together by introducing topological defects, e.g. pentagonal and heptagonal rings, into perfect hexagonal nanotubes to generate junctions. In terms of the SC nanotruss, each junction connects 6 SWCNTs covalently, and consequently, a 6-terminal junction is named (Fig. 1b and c). There are 8 three-tube-constructed surfaces in each junction with three heptagons in every surface, and thus 24 heptagons are presented in total (colored in green). This SC nanotruss has the same architecture of the one considered in Refs. [23] and [47]. Furthermore, the FCC nanotruss consists of 12 terminals per junction, i.e. a 12-terminal junction (Fig. 1g and h). The defects in this type of junction can be categorized into two parts: (1) eight

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three-tube-constructed surfaces with three heptagons per surface, similar to the 6-terminal junction (colored in green) and (2) six four-tube-constructed surfaces with six heptagons per surface (colored in red), so the 12-terminal junction contains 60 heptagons in total. It should be noted that these junctions can have a variety of configurations, as long as the number of non-hexagonal rings obeys Euler's law for polygon [49] and each carbon atom has three covalent bonds (see the Supporting Information S1 for details). Different junction configurations can give rise to changes in mechanical behaviors [21]. In this work, however, we only employ the junctions described in Fig. 1, and all the nanotubes are (8,8) armchair carbon nanotubes. We focus on the dependence of thermal and mechanical properties on the architecture rather than the junctions, because compared with the junction configurations, the architecture has a greater influence on the overall properties of these nano-architected metamaterials [33]. Although, more technological advancements are needed to synthesize 3D CNT networks as ideal as Fig. 1, our study can still explore the properties of nanoarchitected materials and shed lights on the fabrication and application of these novel nanomaterials. Advances in 3D printing technology, e.g. direct-laser-writing optical lithography [50], have also made it possible to fabricate nanoscale materials with complex architectures. MD simulation is performed on six sets of SC and FCC nanotrusses with different SWCNTs lengths. For further discussion, we distinguish our systems in terms of “SC-N” and “FCC-N” terminology, where N represents the number of the hexagon rings in each nanotube connecting adjacent junctions, as can be seen in Fig. 1d,i. Six different tube-length-parameters (N ¼ 3; 7; 12; 16; 20; 24) are chosen in this study corresponding to a wide range of tube length. The atom number and density of SC-N and FCC-N unit cells are listed in Supporting Information S2. As the tube length parameter N increases from 3 to 24, the density of the FCC nanotrusses decreases from 0:82g=cm3 to 0:23g=cm3 , which is much smaller than that of SWCNTs (rSWCNTs z2:27g=cm3 ).

2.2. Computational method Thermal and mechanical properties of the SC and FCC nanotrusses are simulated using LAMMPS package [51] with AIREBO potential [52]. Applying a reverse non-equilibrium molecular (rNEMD) algorithm of Muller-Plathe [53], thermal conductivities are calculated, and mechanical properties are simulated by utilizing the quasi-static displacement-controlled deformation method. Thermal properties are simulated along the [100] direction; mechanical properties are simulated along the [100], [110], and [111] directions. These three directions are represented by solid black arrows in Fig. 1e. Owing to the symmetry of the unit cells, the [100], [010], and [001] directions are equivalent; the [110], [101], and [011] directions are equivalent. Details of thermal and mechanical simulation are documented in Supporting Information S3. To verify our simulation procedure, the properties of a single ð8; 8Þ SWCNT are simulated using the same aforementioned methods and compared with previous studies (Supporting Information S4). It should, however, be noted that bond-breaking behavior is hard to be precisely captured with empirical potential functions used in MD simulation [54] and it is generally accepted that fracture strain and fracture strength calculated from MD simulation are less accurate than for Young's modulus obtained near the equilibrium state [25,54]. Nevertheless, the fracture patterns of CNT are still in accordance with the experimental findings [25], and therefore all the fracture behavior observed in this paper are qualitatively analyzed.

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Fig. 1. Schematic of the SC and FCC nanotrusses: Unit cells of the (a) SC and (f) FCC nanotrusses, with the tube length parameter N ¼ 12. (Colors of the atoms represent the x coordinate and are only used for visualization purpose.) (bec) Different perspectives of the 6-terminal junction, which connects (8,8) SWCNTs covalently to form the SC nanotruss. Twenty-four heptagons are colored in green. (geh) Different perspectives of the 12-terminal junction, which connects (8,8) SWCNTs covalently to form the FCC nanotruss. Twentyfour heptagons are colored in green, and the other thirty-six heptagons are colored in red. Two (d) 6-terminal and (i) 12-terminal junctions are connected by an (8,8) SWCNT, with the tube length parameter N ¼ 3. Schematic of the transversal surfaces for the (e) SC-12 and (j) FCC-12 nanotrusses. The grey sections represent the max-area, and the purple sections represent the atomic-area. All atomistic pictures in this paper are generated by open visualization tool OVITO package [48]. (A colour version of this figure can be viewed online.)

3. Results and discussion 3.1. Low thermal conductivity In order to obtain the thermal conductivity of nano-architected materials, we use the following equation: k ¼ J=VT, where heat flux J is calculated by the equation J ¼ 2dQtA, in which Q ; dt, and A are, respectively, the accumulative total heat flux, the accumulative time, and the cross-section area. Q is divide by 2 since heat flux goes in two opposite directions due to periodic boundary conditions. There are two definitions corresponding to the cross-section area: the atomic-area and the max-area. The atomic-area (colored in purple, Fig. 1e,j) only contains the cut section of the tubes and junctions excluding voids, while the max area (colored in grey, Fig. 1e,j) represents the entire cross-section area of the unit cell. Following the convention, the thickness of the nanotube is taken to be the interlayer distance of two graphene sheets, i.e. 3.4 Å [7]. In this work, the cross section is selected as shown in Fig. 1e,j, and the atomic-area of the SC and FCC nanotrusses are 463.49 Å2 and 1127.16 Å2, respectively. Atomic-area does not change with the tube length and is often used in previous studies to calculate the thermal conductivity of nanotube. With atomic-area, one can explore the phonon scattering mechanism of the nanotrusses. By contrast, the max-area rises quadratically with the increase in tube length; such area is commonly used for material design. In this study, both areas are computed and analyzed for the calculation of thermal conductivity, yet the max-area is highlighted because a novel nanoarchitected material is presented here. Temperature profiles of the SC-3, FCC-3, SC-16 and FCC-16 nanotrusses are presented in Fig. 2a. The solid black lines are linear fits of the central parts of the temperature profile, where temperature gradients can be obtained. The inset shows the accumulative heat flow in these nanotrusses. For both nanotrusses, an

increase in tube length leads to a decrease in temperature gradients and a rise in energy exchange. This suggests kAtomicarea will rise with the tube length as listed in Table 1. The reason can be found in Fig. 2c; for both of the SC and FCC nanotrusses, temperature gradients circled by red dashed ellipses are much higher than other parts which indicate lower local thermal conductivity at these parts. These parts (circled by red dashed ellipses in Fig. 2c) are associated with the junctions in nanotrusses, which are major sources of phonon scattering. Junction density decreases with tube length; as a result, heat better transports in nanotrusses with longer constituent tubes. On the contrary, kMaxarea is inversely proportional with the tube length (Table 1). There are two competitive factors influencing the thermal conductivity: (1) phonon scattering rate decreases with the increase of the tube length, which will increase the thermal conductivity, and (2) the max-area increase quadratically with the tube length, which will decrease the thermal conductivity and is the dominating factor. The thermal conductivities of the SC nanotrusses calculated in this work are in line with the results presented in Ref. [23] (with less than 5% error). Comparing k of the SC nanotruss with the FCC nanotruss in Table 1, one can find that the FCC nanotrusses have much smaller kAtomicarea than the SC nanotrusses for all the tube lengths, which is due to the increased phonon scattering in the 12-terminal junctions of FCC nanotruss. Additionally, the heat conduction direction of the FCC nanotrusses is not the tube axial direction, which is not the case for the SC nanotruss. Considering max-area, the FCC nanotrusses still have a smaller kMaxarea than the SC nanotrusses when the tube length is small. Nonetheless, when N  16, the FCC nanotrusses have larger kMaxarea than the SC nanotrusses because FCC nanotrusses possess much larger atomic-area (much large density) to conduct heat. Fig. 2b shows thermal conductivity of the SC and FCC

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Fig. 2. (a) Steady state temperature profiles of the SC-3, SC-16, FCC-3 and FCC-16 nanotrusses as a function of the x coordinate (one unit cell is repeated 10 times in the x direction). The solid black lines are linear fits of the central parts of the temperature profile to obtain temperature gradients. The inset shows the accumulative heat flux in these nanotrusses. (b) The thermal conductivity k in the [100] direction of SC and FCC nanotrusses, calculated with the atomic-area (black) and the max-area (red), changes with the density r of the nanotruss. (c1-c2) Temperature profiles (black lines and points) of the SC-16 and FCC-16 nanotrusses extracted from 10 repeated unit cells in the thermal simulation. The temperature is the average temperature of bins as described in the Supporting Information S3. The red dashed ellipses circled the parts that temperature gradient is larger than 1.5K=Å and 2.0K=Å for SC and FCC nanotruss, respectively. (Colors of the atoms represent the x coordinate and are only used for visualization purpose.). (A colour version of this figure can be viewed online.)

Table 1 Thermal conductivity of the SC-N and FCC-N nanotrusses calculated by the atomic-area and the max-area. N

3

7

12

16

20

24

SC-N kAtomicarea ðW=ðm,KÞÞ FCC-N kAtomicarea ðW=ðm,KÞÞ SC-N kMaxarea ðW=ðm,KÞÞ FCC-N kMaxarea ðW=ðm,KÞÞ

17.92 7.28 4.62 3.31

27.08 10.13 3.28 2.84

35.74 11.74 2.23 2.04

38.54 12.96 1.60 1.64

42.29 14.40 1.25 1.39

47.92 16.44 1.06 1.24

nanotrusses changes with the system density r. For both of the atomic-area and the max-area, the FCC nanotrusses always have smaller thermal conductivities than the SC nanotrusses at a fixed density; kMaxarea is between 1:24W=ðm,KÞ to 3:31W=ðm,KÞ for the FCC nanotrusses. Consequently, the FCC nanotruss is a better thermal insulator compared with the SC nanotruss. Most of the common thermal insulators with low thermal conductivity are not mechanically strong limiting their multifunctional application. As a result, it is important to also evaluate the mechanical modulus and strength of the developed nano-architected advanced materials. 3.2. High specific modulus in all directions Fig. 3a displays Young's modulus E of the SC and FCC

nanotrusses for different tube length parameters N in [100], [110], and [111] directions. Young's modulus is calculated by the linear fit of the strain-stress curve in the strain range of 0  5%. In stress calculations, the cross-section area is taken to be the current maxarea. The dashed and solid black lines illustrate that the FCC nanotrusses always have greater Young's modulus (E) than the SC nanotrusses in the [100] direction (Fig. 3a). Comparing the other two directions, the FCC nanotrusses have smaller Young's modulus than the SC nanotrusses only for tube length parameters N smaller than 6 (N < 6), represented by a blue phase in Fig. 3a. The reason is associated with the fact that the tube length is even smaller than the junction diameter for N < 6; as a result, the structural properties are dominated by the junctions rather than the architecture for N < 6. Apparently, the E values of the SC nanotrusses drop

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Fig. 3. (a) Young's modulus E of SC (dashed lines) and FCC (solid lines) nanotrusses with different tube length parameters N in [100] (black lines), [110] (blue lines), and [111] (red lines) directions. (b) The specific modulus M and the thermal conductivity k of the FCC nanotrusses ([100] direction), FCC diamond-trusses ([100] direction), technical ceramics [55], metals [55], polymers [55], foams [55], non-technical ceramics [55], composites [55], and other carbon materials (carbon nanotube [3,56e58], graphene [7,59,60], and diamond [61e63]). (A colour version of this figure can be viewed online.)

dramatically nearly to zero in [110] and [111] directions with the rise of tube length since there is no tube capable of standing the applied stress and the nanotruss is very flexible. On the contrary, the FCC nanotrusses can stand stress in all these three directions and Young's modulus of FCC nanotrusses in all three directions converge to the same value by increasing the tube length. Hence, the FCC nanotrusses are much stronger than the SC nanotrusses. Furthermore, Young's modulus of the FCC nanotrusses changes in a wide range from 11:66GPa to 93:32GPa with the material density changes from 0:23g=cm3 to 0:82g=cm3 , which enables E to be adjusted depending on specific requirements in the design of lightweight multifunctional materials. The E values of FCC nanotrusses are about one or two orders smaller than the E ¼ 1047:81GPa of a single (8,8) SWCNT, due to the large hollow areas in the FCC nanotrusses. However, the specific modulus M (M ¼ E=r) of FCC nanotrusses is in the range of 49:79MN,m=kg  114:15MN,m=kg, which is almost in the same order of the specific modulus 461:59MN,m=kg of a single (8,8) SWCNT (Supporting Information S5). 3.3. Comparison with other materials The specific modulus M and the thermal conductivity k of FCC nanotrusses, FCC diamond-trusses, technical ceramics [55], metals [55], polymers [55], foams [55], non-technical ceramics [55], composites [55], and other carbon materials (carbon nanotube [3,56e58], graphene [7,59,60], diamond [61,62]) are provided in Fig. 3b. These materials cover almost all engineering materials. Although only [100] direction specific modulus for FCC nanotrusses is presented in Fig. 3b, it has been shown in Fig. 3a the specific modulus in the other two directions is not far away from the [100] direction. The maximum and minimum k and M of technical ceramics, metals, polymers, foams, non-technical ceramics, composites are extracted from Ashby charts [55]. FCC diamond-trusses are the macroscopic counterparts of the FCC nanotrusses, whose thermo-mechanical properties are predicted in a continuum level using standard mechanics homogenization method as described in the subsection “macroscopic counterparts”. The blue shaded domain in Fig. 3b has high specific modulus (M  40MN,m=kg) and low thermal conductivity (k  5W=ðm,KÞ). Only the FCC nanotrusses and a part of composites are in the blue domain, but the FCC nanotrusses have lots of other advantages over pristine composites,

such as lightweight, high thermal stability and some extraordinary electronic properties. Consequently, the FCC nanotrusses can be utilized as excellent mechanically robust thermal insulators. FCC diamond-trusses have similar specific modulus, but much larger thermal conductivity compared with FCC nanotrusses, due to the different thermal conduction mechanism in different scales and the effect of size-dependency on the thermo-mechanical properties of nano-architected materials.

3.4. High thermal stability As a thermal insulating material, the nanotruss may work in high-temperature environments. Therefore, thermal and mechanical simulation are repeated at 1000K, 2000K and 3000K temperature. Table 2 shows that k decreases with temperature increase, which indicates that the nanotruss is still an outstanding thermal insulator at high temperature. As for E, from 300K to 3000K, Young's modulus only decreases 10:80% and 13:93% for the SC-3 and FCC-3 nanotrusses, respectively. It is worth mentioning that 3000K is an extremely high temperature, at which most materials have already achieved their melting points (the melting points of aluminum, copper, and silicon are 933K [64], 1358K [64], and 1687K [64], respectively), whereas, FCC nanotrusses still work without any damage in 3000K. In experiments, two adjacent carbon nanotubes can occasionally coalesce into one nanotube [65e67] at temperature higher than 1000K; a phenomenon which does not happen in our carbon nanotube-based nanotrusses even at 3000K. The reason is the coalescence conditions (such as vacancy, specific junction topology, and enough space [65e67]) are not satisfied in our nanotrusses. Required experimental technology for synthesizing of our proposed FCC carbon nanotube-based

Table 2 Thermal conductivities kmaxarea and Young's modulus E of the SC-3 and FCC-3 nanotrusses at different temperature. Temperature (K)

SC-3k (W/m,K)

FCC-3k (W/m,K)

SC-3E (GPa)

FCC-3E (GPa)

300 1000 2000 3000

4.62 3.56 3.14 2.76

3.31 2.97 2.69 2.41

71.20 68.57 64.09 63.51

93.32 89.87 84.88 80.32

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nanotrusses should avoid exposing the nanotrusses to electron irradiation since it can induce vacancies causing coalescence at high temperature. To show the stability of our carbon nanotubebased nanotrusses, we conduct another simulation on the FCC-7 unit cell in an NVT ensemble under ambient pressure at 3000K and 4000K, respectively (Supporting Information S8). There is no melting or coalescence at 3000K. When temperature reaches 4000K at 364ps, the carbon-based nanotrusses starts to melt, which is aligned with the reported melting point of nanotubes that is between 3000K and 4000K [68,69]. Hence, FCC nanotrusses are a fantastic mechanically robust thermal insulator which can presume their multifunctionality for a wide range of temperature and can be applied to extreme conditions, such as high-temperature vacuum aerospace environments. 3.5. High fracture strain and fracture strength The stress-strain curve of the FCC-12 nanotruss in the [100] direction with different simulation conditions can be found in Fig. 4a. In this study, all tensile tests are conducted with a strain rate v ¼ 0:001=ps, and one three-dimensional (3D) unit cell is repeated twice in all three directions (solid black line). Since there is an abrupt change in the curve at point B, we compress the system before and after point B (the purple circles and the green triangles). The stress-strain curve goes the same way as the tensile test only when we compress the nanotrusses before point B, which means plastic deformation occurs at point B and point B is the initial fracture point. When the strain rate decreases to 0:0005=ps and the unit cell repeats in all directions for three times, all the stress-strain curves coincide before point C (Fig. 4a), which suggests Young's modulus, initial fracture strain and initial fracture strength are accurate and strain rate independent. Nevertheless, the ultimate strength (point D) differs for alternative strain rates and different repetition of unit cells. The difference between the ultimate strength in the three curves is less than 10%. As discussed previously, the bond-breaking behavior is hard to be captured precisely with empirical potential functions applied in MD simulations [54]. The nanotruss rupture point here (point E), changes in a wide range

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for different simulation conditions, that is why we do not report the rupture point in this paper. Fig. 4b illustrates the corresponding snapshots of the nanotruss during uniaxial tensile tests. The letters on the pictures correspond to the points marked in the stress-strain curve. The atom color represents the potential energy of each atom. It can be seen that the atoms in the junctions have higher potential energy than other atoms, indicating stress concentration and potential development of plastic deformation in the junctions of the nano-architected materials. Fig. 5a shows stress-strain curves of the SC-12 and FCC-12 nanotrusses in the [100], [110], [111] directions and illustrates the deformation mechanisms of SC and FCC nanotrusses. Most FCC nanotrusses undergo an initial fracture point, then the stress increase continuously or fluctuates till the rupture point. The fluctuations represent the release of energy caused by the break of covalent bonds. Conversely, the SC nanotrusses undergo an elastic deformation till the rupture point in the [110] and [111] directions. As a result, the initial fracture point of SC nanotruss coincides with the ultimate strength point, since there is no tube which can stand the applied stress directly in [110] and [111] directions and thus SC nanotrusses are very flexible. Moreover, the stress distributes relatively uniformly on all the atoms till the rupture point, as can be seen in Fig. 5b for SC-12 in [110] and [111] directions. This different deformation mechanism results in larger initial fracture strain, initial fracture strength and ultimate strength for the SC nanotrusses in the [110] and [111] directions, as shown in Supporting Information S6. Furthermore, for FCC nanotrusses, the initial fracture strain is between 15:2% and 34:2%, the initial fracture strength is between 1:65GPa and 28:31GPa, and the ultimate strength is between 4:27GPa and 33:03GPa (Supporting Information S6), which are much larger than the fracture strain (9:8%  11:1%) and tensile strength (5:6GPa  8:3GPa) of the graphene-based FCC nanotrusses reported in Ref. [46]. The animations of the uniaxial tensile tests of these nanotrusses are provided in Supporting Information Movie 1e6. Supplementary video related to this article can be found at https://doi.org/10.1016/j.carbon.2018.01.082.

Fig. 4. (a) Stress-strain curve of the FCC-12 nanotruss in the [100] direction with different simulations conditions. The solid black line represents a strain rate v ¼ 0:001=ps and one unit cell is repeated twice in all three directions. The dashed red line represents a strain rate v ¼ 0:0005=ps and one unit cell is repeated twice in all three directions. The dotted blue line represents a strain rate v ¼ 0:001=ps and one unit cell is repeated three times in all three directions. The purple circles and the green triangles are the stress-strain points in the compression tests before and after the initial fracture point B, respectively. (b) Snapshots of the nanotrusses during tensile tests. The letters on the pictures correspond to the points marked in the stress-strain curve. The atom color represents the potential energy of each atom. (A colour version of this figure can be viewed online.)

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Fig. 5. (a) Stress-strain curve of the SC-12 and FCC-12 nanotrusses in the [100], [110], [111] directions. (b) Snapshots of these nanotruss in the tensile tests at the initial fracture point. The atom color represents the potential energy of each atom. (A colour version of this figure can be viewed online.)

3.6. Macroscopic counterparts Apart from thermal and mechanical properties of SC and FCC nanotrusses presented above, thermal and mechanical properties of SC and FCC architected periodic cellular materials (metamaterials [16,17]) in macro scale are explored to provide a comprehensive understanding on the response of similar

architecture in multiple scales. On macro scale, we build SC and FCC architected materials using diamond instead of carbon nanotube, both of which are made of carbon and have extremely large thermal conductivity and Young's modulus. Geometries of SC and FCC diamond-trusses (Fig. 6a) are the same as SC and FCC nanotrusses, respectively. In analogy with SC and FCC nanotrusses, we also construct SC and FCC diamond-trusses for six different N values. We

Fig. 6. (a) Schematic shapes of SC and FCC diamond-trusses. (b) Thermal conductivity and specific modulus along the (c) [100] and (d) [110] direction of the SC and FCC nanotrusses and diamond-trusses change with relative density. (A colour version of this figure can be viewed online.)

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study their effective mechanical and thermal properties in a continuum scale using a standard mechanics homogenization technique (Supporting Information S7) [70]. The relative dimensions and relative density of the diamond-trusses are chosen to be the same as their counterpart carbon nanotube-based nanotrusses. The relative dimension, including relative tube length, tube diameter, tube thickness, junction diameter, and junction thickness, refers to the real dimensions divided by the unit cell length. Similarly, the relative density of the diamond-truss and nanotruss refers to their material density divided by the density of diamond and CNTs, respectively. As shown in Fig. 6bed, homogenized results reveal that macroscopic counterparts have similar thermal (Fig. 6b, at the same relative density, FCC architecture always has a smaller effective thermal conductivity than SC architecture for both nanotruss and diamond-truss. The effective thermal conductivity of diamondtruss, however, is about two orders larger than the thermal conductivity of nanotruss due to different phonon scattering mechanism in micro and macro scale.) and mechanical (Fig. 6c and d, at the same relative density, compared with the SC architecture, the FCC architecture has a smaller specific modulus before the intersection point and a bigger specific modulus after the intersection point for both nanotruss and diamond-truss) behavior to SC and FCC nanotrusses, which sheds lights on how the concept of “architected” materials can be used in multiple scales from micro to macro scale. 4. Conclusions In conclusion, we systematically explore thermal and mechanical properties of the carbon nanotube-based FCC nanotruss as a new category of nano-architected metamaterials using MD simulation. The numerical observation implies that FCC carbon nanotube-based nanotrusses are a new category of lightweight, mechanically robust, and thermally insulative nano-architected metamaterials with high thermal stability. These nanoarchitected metamaterials with tunable thermo-mechanical properties are superior to existing mechanically robust thermal insulators and can be used for many applications, such as thermal protector walls for extreme conditions [71], electronic devices [33,72,73], and thermal management [38,39]. Thermal and mechanical properties of their macroscopic counterparts are also evaluated by using multiscale standard mechanics homogenization which corroborates the scale independency of the unprecedented properties found for the FCC truss-like architectures and provides a comprehensive understanding on the response of similar architectures in multiple length scales. Our findings shall guide the design and manufacturing of next-generation multifunctional cellular solids and structures using additive manufacturing technology. Acknowledgements The work was funded by Peking University Short-term Visiting Program, by McGill University and Natural Sciences and Engineering Research Council of Canada (NSERC) through NSERC Discovery Grant RGPIN-2016-0471, and by NSFC (Grant No. 11521202). Appendix A. Supplementary data Supplementary data related to this article can be found at https://doi.org/10.1016/j.carbon.2018.01.082. References [1] S. Iijima, Synthetic nano-scale fibrous matrix, Nature 56 (1991) 354e358.

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